From the actions on the \(\mathcal {F}_{i}\)  and the \(\mathcal {H}_{i}\) ,
composition by way of \(\mathrm {aut}\mathcal {N}P/\mathbf {F}_{P}^{\#}\) 
produces an automorphism of \(\mathrm {Sym}(6)\) . For instance, the
transposition \(\left( 12\right) \)  on the \(\mathcal {F}_{i}\)  effects \((15)(26)(34)\)  on the \(\mathcal {H}_{i}\) , and \((123)\)  produces \((164)(253)\) .
The automorphism is indeed outer. The enhanced notation shows that a
duad-syntheme correspondence, suggested by J. J. Sylvester and described in
[1]}, is present in (REF ). Each edge appears in two rows, and
matching the row index pair with the edge sets up such a correspondence. For
example, \(\mathbf {12}\longleftrightarrow \mathbf {162345}\)  and \(\mathbf {36}\longleftrightarrow \mathbf {142356}\) . An outer automorphism also provides a
correspondence, matching transpositions with triple transpositions. For more
on this topic, see [2]}.
